Harmonie Force Fields and Bond Orders for Naphthalene, Anthracene, Biphenylene and Perylene with Mean Amplitudes for Perylene J. C. Whitmer *, S. J. Cyvin and B. N. Cyvin Division of Physical Chemistry, The University of Trondheim, N-7034 Trondheim-NTH, Norway Z. Naturforsch. 33a, 45-54 (1978); received October 25, 1977 Complete normal coordinate analyses were performed for naphthalene, anthracene, biphenylene and perylene, starting from a simple force field with seven adjustable force constants. A rela- tionship between bond orders and carbon-carbon stretching force constants was deduced from: (a) bond distances as a function of bond orders, (b) a version of Badger's rule relating stretching force constants to the bond distances. The relationship was used to modify the initial seven- parameter force field, and the vibrational frequencies calculated from both the initial and modi- fied force fields are discussed. In general the simple force field approximation produces sets of frequencies in remarkably good agreement with experimental assignments. The force field ap- proximation failed badly when applied to benzene. No obvious explanation was found for this unexpected feature, which makes it worth while to continue the investigations. The mean ampli- tudes of vibration were calculated. For perylene an account of the complete set of mean ampli- tudes is given for the first time. Introduction molecule (thirty-two atoms) for which the calculated mean amplitudes are given here for the first time. Conjugated systems in organic chemistry have been studied by many investigators. The Hiickel Bond Orders and Interatomic Carbon-Carbon Molecular Orbital (HMO) theory [1] has proved to Distances be a simple, but efficient, tool in these studies. In the present work we attempt to combine the HMO The bond order (P) for conjugated systems is theory with vibrational normal coordinate analysis defined here in the usual way [1] as numbers [2] for some conjugated systems. The main purpose O^P^l. (1) is to correlate bond orders from the simple HMO The definition implies P= 1.000 for the "pure" theory with force constants of carbon-carbon double bond in ethylene. For benzene and graphite stretchings. In addition we are interested in the the bond orders are 0.667 and 0.525, respectively. calculated mean amplitudes of vibration [3], Table 1 shows the bond orders calculated here for especially for the carbon-carbon distances. the four molecules considered. The results for The four molecules selected for the present naphthalene, anthracene and biphenylene are investigation are: naphthalene CioHs, anthracene identical with those of Heilbronner and Bock [1], C14H10, biphenylene C^Hg and perylene C20H12. while perylene is not included in their compilations. The structures for all of them have been investigated However, also for perylene the bond orders have by gas electron diffraction, and in one case (bi- been calculated by many investigators; the phenylene) observed mean amplitudes are reported. literature is too voluminous to be cited here. In all cases the mean amplitudes calculated from Figure 1 shows that the conventional representa- spectroscopic data would be of great help in the tions of aromatic structures with simple and double interpretation of the electron diffraction experi- bonds are highly misleading when resonance ments. The four molecules have also been in- structures are not taken into account. vestigated spectroscopically, and more or less Many attempts have been done to correlate the complete assignments of vibrational frequencies bond orders with interatomic CC distances in have been proposed. Perylene is a relatively large conjugated systems of organic molecules. One of the most successful approaches is due to Coulson * Permanent address: Department of Chemistry, Western Washington University, Bellingham, Washington, USA- [4] and based on molecular orbital theory. The 98225. relatively simple formula Reprint requests to be sent to Prof. S. J. Cyvin, Division 0.192 P of Physical Chemistry, The University of Trondheim, N-7034 Trondheim-NTH, Norway. r(A) = 1536 " P + 0.765(1 --Pf <2> 46 J. C. Whitmer, S. J. Cyvin, and B. N. Cyvin • Vibrations of Condensed Aromatics reproduces fairly well many experimental CC Table 1. Bond orders with calculated and observed inter- distances in organic molecules. Here 1.536 Ä is the atomic CC distances. "pure" single bond distance in ethane (correspond- Distance Calculated Observed13 ing to P = 0.000), while 0.192 Ä is the difference typea Bond Distance Distance between the single and double bond distances. order [A] [A] Table 1 also includes the calculated CC bond distances according to the Coulson formula (2). Naphthalene a 0.603 1.408 1.412 They are seen to compare well with the observed b 0.725 1.387 1.371 distances. The quoted values for naphthalene [5], c 0.555 1.417 1.422 anthracene [5], biphenylene [6] and perylene [7] d 0.518 1.424 1.420 are from the cited gas electron diffraction works. Anthracene a 0.586 1.411 1.419 The experimental data are in very good agreement b 0.737 1.385 1.390 with x-ray results for naphthalene [8], anthracene c 0.535 - 1.421 1.420 [8] and biphenylene [9]. Most of the differences d 0.485 1.430 1.425 e 0.606 1.408 1.404 between the calculated and observed distances are Biphenylene between 0.01 and 0.02 Ä. Somewhat more pro- a 0.691 1.393 1.370 nounced discrepancies (differences of 0.04—0.05 Ä) b 0.621 1.405 1.428 occur for the weakest bonds in biphenylene and c 0.683 1.394 1.372 d 0.565 1.415 1.432 perylene. The high experimental value for distance e e 0.263 1.475 1.524 in biphenylene (see Table 1) is supported by the Perylene x-ray result (1.514 Ä) [9]. For the distance e in a 0.629 1.404 1.420 perylene (the peri bond) the quoted value (Table 1) b 0.707 1.390 1.375 b' 0.644 1.401 1.402 falls in-between another electron diffraction c 0.552 1.417 1.412 result (1.493 Ä) [10] and an x-ray value (1.471 Ä) e' 0.529 1.422 1.429 [11] published the same year. d 0.526 1.422 1.451 e 0.414 1.444 1.483 a See Fig. 1. Structural Parameters 13 From gas electron diffraction [5—7]. In the present normal coordinate analysis we adherred to the structural parameters for naphtha- lene and anthracene used in previous normal coordinate analyses [12, 13]. They include the CC distances from the x-ray investigation of Cruickshank and Sparks [8], which are practically identical to the electron diffraction values quoted in Table 1. The structural data used for biphenylene are from Yokozeki et al. [6], from whom the CC distances are quoted in Table 1. Also for perylene the gas electron diffraction data were adopted. Dallinga et al. [7] have given an over-determined set of structural parameters. We calculated the BIPHENYLENE distance a (cf. Fig. 1 and Table 1) from the other CC distances and reported CCC angles and obtained 1.421 Ä, which is practically the same as the electron diffraction value contained in Table 1. A Simple Force Field for Aromatics Several preliminary calculations were performed PERYLENE in order to obtain a simple approximate force field Fig. 1. Naphthalene, anthracene, biphenylene and pery- lene : Conventional drawings with single and double bonds, transferable to many kinds of aromatic molecules. and drawings with relevance to calculated bond orders. 47 J. C. Whitmer, S. J. Cyvin, and B. N. Cyvin • Vibrations of Condensed Aromatics A harmonic force field was developed with only Table 3. Calculated and observed frequencies (cm-1) for seven adjustable parameters. It may be expressed naphthalene. in terms of a diagonal F matrix based on valence Species Calculated coordinates including redundancies. Numerical a values for these parameters were initially obtained Approx. Refined for Observed CC stretchings to give good agreement between calculated and observed frequencies in naphthalene. Table 2 shows Ag 3036 3036 3060 3030 3030 3031 1564 1559 1577 1469 1456 1460 Table 2. Force constants of the approximate diagonal 1366 1363 1376 F matrix. 1057 1062 1145 868 873 1025 Type mdyne/A 674 663 758 426 423 512 CC stretching 4.7 Big 934 934 943 CH stretching 5.0 702 702 717 CCC bending 0.7 312 312 386 CCH bending 0.3 CCCC torsion 0.1 Big 1305 1305 980 CH out-of-plane bend 0.2 952 952 876 CC out-of-plane bend 0.15 679 679 846 379 379 461 B3g 3039 3039 3092 the seven different types of valence coordinates and 3031 3031 3060 1731 1721 1624 the final numerical values of the respective force 1388 1399 1438 constants. The angle coordinates (bendings and 1244 1244 1239 torsions) were scaled with the appropriate bond 1050 1050 1158 931 945 936 distances so that all force constants have the same 534 531 506 units (md}me/A). The CCC bendings are in typical Au 1150 1150 970 cases analogous to the ring bendings in benzene; 776 776 841 only the inner angles for each ring are included in 417 417 581 185 185 195 the more complex aromatics. The torsions intro- 3040 3040 duced are all supposed to be of the CCCC "boat" Bin 3065 3031 3032 3058 type as in benzene itself. For the CC out-of-plane 1489 1509 1595 bendings (which do not exist in benzene itself) one 1328 1324 1389 1191 1187 1265 should tend to include a minimum number of such 980 993 1125 coordinates. More detailed accounts on the sets of 803 795 747 valence coordinates chosen in the different cases 284 285 359 are given under the descriptions of the individual Bzu 3035 3035 3090 3030 3030 3027 molecules.